The masses of star clusters range over seven decades , from ten up to one hundred million solar masses .
Remarkably , clusters with masses in the range 10 ^ { 4 } M _ { \odot } to 10 ^ { 6 } M _ { \odot } show no systematic variation of radius with mass .
However , recent observations have shown that clusters with M _ { cl } \gtrsim 3 \times 10 ^ { 6 } M _ { \odot } do show an increase in size with increasing mass .
We point out that clusters with M _ { cl } \gtrsim 10 ^ { 6 } M _ { \odot } were optically thick to far infrared radiation when they formed , and explore the hypothesis that the size of clusters with M _ { cl } \gtrsim 3 \times 10 ^ { 6 } M _ { \odot } is set by a balance between accretion powered radiation pressure and gravity when the clusters formed , yielding a mass-radius relation r _ { cl } \sim 0.3 ( M _ { cl } / 10 ^ { 6 } M _ { \odot } ) ^ { 3 / 5 } { pc } .
We show that the Jeans mass in optically thick objects increases systematically with cluster mass .
We argue , by assuming that the break in the stellar initial mass function is set by the Jeans mass , that optically thick clusters are born with top heavy initial mass functions ; it follows that they are over-luminous compared to optically thin clusters when young , and have a higher mass to light ratio \Upsilon _ { V } = M _ { cl } / L _ { V } when older than \sim 1 Gyr .
Old , optically thick clusters have \Upsilon _ { V } \sim M _ { cl } ^ { 0.1 - 0.3 } .
It follows that L _ { V } \sim \sigma ^ { \beta } , where \sigma is the cluster velocity dispersion , and 3.6 \lesssim \beta \lesssim 4.5 .
It appears that \Upsilon _ { V } is an increasing function of cluster mass for compact clusters and ultra-compact dwarf galaxies .
We show that this is unlikely to be due to the presence of non-baryonic dark matter , by comparing clusters to Milky Way satellite galaxies , which are dark matter dominated .
The satellite galaxies appear to have a fixed mass inside a fiducial radius , M ( r = r _ { 0 } ) = const . .